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Theorem nelbrnel 42915
 Description: A set is related to another set by the negated membership relation iff it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
Assertion
Ref Expression
nelbrnel ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵𝐴𝐵))

Proof of Theorem nelbrnel
StepHypRef Expression
1 nelbr 42913 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
2 df-nel 3067 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
31, 2syl6bbr 281 1 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∈ wcel 2051   ∉ wnel 3066   class class class wbr 4925   _∉ cnelbr 42910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-nel 3067  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-nelbr 42911 This theorem is referenced by: (None)
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